Stable and unstable molecules under supercritical and cryogenic conditions

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Stable and unstable molecules under supercritical and cryogenic conditions
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    Abstract
        Page vi
    Chapter 1. Electron spin resonance (ESR) theory
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Chapter 2. The nature of a spin probe under the influence of supercritical carbon dioxide (CO2)
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
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        Page 30
        Page 31
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        Page 33
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        Page 48
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    Chapter 3. Vibrational spectroscopy theory
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
    Chapter 4. Infrared spectra of Nb12C, Nb13C and NbO2 molecules matrix isolated in rare gas matrices
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
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        Page 69
        Page 70
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        Page 72
        Page 73
    Chapter 5. Densimeter
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
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        Page 115
        Page 116
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        Page 118
        Page 119
        Page 120
        Page 121
    References
        Page 122
        Page 123
        Page 124
    Biographical sketch
        Page 125
        Page 126
        Page 127
        Page 128
Full Text












STABLE AND UNSTABLE MOLECULES UNDER SUPERCRITICAL
AND CRYOGENIC CONDITIONS













By

TROY D. HALVORSEN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1998















ACKNOWLEDGEMENTS

The author wishes to thank first and foremost his family whose love and support

have fostered his growth as an individual and has propelled him to this position in his

life. His mother (Susan) who personally sacrificed and was unwilling to let the author

accept anything but the best. His dad (Joe) who first introduced the author to academia

and has "gently" guided his path towards higher education. The author does not believe

that his dad would have ever believed in his wildest dreams that he would eventually

accept an assistant professorship at Truman State University. When his dad dropped him

off at Illinois State University in the fall of 1986 to begin his undergraduate career as a

chemistry major, the author believes that his exact words were "I think your strengths are

in Sociology."

The author would also like to acknowledge his wife. Kim, for her love and

support. Kim almost did not follow the author to Gainesville. but the author cannot

imagine life without her and their new baby girl, Alana Marie ("Peaches"), who has been

a divine gift.

The author is deeply indebted to Dr. William Weltner, Jr., who unquestioningly

accepted the author into his group with only two years remaining in his Ph.D. work. The

author thanks him for giving him the opportunity, and it cannot go without saying that

Dr. Welter's striving for the truth has made an indelible impression.








The author would also like to extend his thanks to Dr. Sam Colgate for allowing

the him to work in his laboratory and for passing on to him his vast knowledge and skills.

Thank you.

The author cannot say enough about the wonderful support staff at the University

of Florida. especially the machine shop personnel. The projects the author has worked on

would not have come to fruition without the skill and knowledge of Joe Shalosky, Todd

Prox, and Mike Herlevich. We are truly spoiled to have access to these people.

Thanks also goes out to the author's cohorts and friends at UF. Dr. Aaron

Williams. Dr. Heather Weimer, Dr. John Graham. Johnny Evans, and especially Dr.

Richard Van Zee who might be the single most impressive scientist I have been

associated with.

The author cannot forget about the "crue" from Illinois State University ("the

Illinois mafia") who will always be apart of the author's life; David E. Kage, Dr. Nick

Kob, Dr. Richard Burton, Dr. Eugene Wagner, Dr. Scott Kassel, and Ben Novak.

The author must also acknowledge the National Science Foundation (NSF) for

financial support.















TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS................................................................................................ ii

A B ST R A C T ....................................................................................................................... vi

CHAPTERS

1 ELECTRON SPIN RESONANCE (ESR) THEORY............................ 1

Classical D description .................................................................... 1
Quantum-Mechanical Treatment.................................................. 2

2 THE NATURE OF A SPIN PROBE UNDER THE
INFLUENCE OF SUPERCRITICAL CARBON DIOXIDE
(C O ,) .................................................................................................1.... 1

General Description of Supercritical Fluids...........................I..... 1I
Introduction................................................................................. 12
Experim ental............................................................................... 14
Results and Discussion ............................................................... 18
C conclusion .................................................................................. 23

3 VIBRATIONAL SPECTROSCOPY THEORY.................................. 50

Classical Description .................................................................. 50
Quantum-Mechanical Treatment................................................ 53

4 INFRARED SPECTRA OF Nb2C, NbsC AND
NbO, MOLECULES MATRIX ISOLATED IN
RARE GAS MATRICES..................................................................... 58
Introduction................................................................................. 58
Experim ental............................................................................... 59
Results and Discussion ............................................................... 60








The Ground State (A3/2) Vibrational Properties of Nb'2C
and Nb C .................................................................................... 67
The N bO, m olecule ..................................................................... 68
Conclusion .................................................................................. 72

5 DEN SIM ETER ...................................................................................... 74

Introduction................................................................................. 74
Therm odynam ic Relationships................................................... 75
First Principles ............................................................................ 78
Design of Densim eter ................................................................. 80
Densim eter Developm ent............................................................ 84
Experim ental............................................................................... 89
Results....................................................................................... 115
Conclusion ................................................................................ 116


REFEREN CES ................................................................................................................122

BIOGRA PHICA L SKETCH ........................................................................................... 125

























V














Abstract of Dissertation Presented to the Graduate School of the University of
Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

STABLE AND UNSTABLE MOLECULES UNDER SUPERCRITICAL AND
CRYOGENIC CONDITIONS

By

Troy D. Halvorsen

August 1998

Chairman: Professor William Weltner, Jr.
Major Department: Chemistry

The electron spin resonance spectra of a spin label (di-tertiary butyl nitroxide,

DTBN) dissolved in supercritical carbon dioxide were obtained. By investigating the

band shapes of the spectra, information about the nature of the solvent can be inferred.

Comparison to simulated spectra provided a means of interpreting these data.

The spectra ofNb2C, Nb3C and NbO, in argon, krypton and neon matrices were

obtained using Fourier Transform infrared spectroscopy (FTIR). The resulting spectra

were assigned through comparison of the isotopic shifts in vibrational frequencies.

A prototype high pressure densimeter was designed, built and successfully tested

with argon and carbon dioxide around ambient temperatures and pressures up to 1800

psia.













CHAPTER 1
ELECTRON SPIN RESONANCE (ESR) THEORY

Classical Description

Classically speaking, an atom may possess electronic orbital motion which creates

a magnetic dipole (p). The description of a dipole moment induced by general angular

momentum is given in equation 1, where PL is the orbital angular momentum, c is the

velocity of light, and e and m are the elementary charge and fundamental mass of an

electron, respectively. The dipole moment is therefore proportional to the magnitude and


S= (im PL ()

direction of the orbital angular momentum (pL) and is scaled by a term more commonly

referred to as the gyromagnetic ratio (y = -e/2mc).

The intrinsic magnetic moment associated with an electron regardless of motion

has been termed "spin." The dipole moment associated with the spin of an electron is

represented in equation 2, and it should be noted that the gyromagnetic ratio has an

additional scaling factor of 2, which is added to account for its anomalous behavior. The

contributions of spin (S) and orbital angular moment (L) can couple to give a resultant

total angular momentum (J) that can interact with an applied or local magnetic field. The

energy of a magnetic dipole that is "perturbed" by a magnetic field is given classically by

equation 3.







2


1 = 2( PS (2)

E = -t H cos 0 (3)

Quantum mechanics has relegated the angle 0 (with respect to the dipole moment

and magnetic field) to discrete space-fixed orientations. There are 2J + 1 orientations of

the total angular momentum (J) with projections of mjh along the magnetic field axis,

where mj can assume values of mj = J, J 1, ... -J. The angular momentum coincident

with the field are therefore the integral multiples of pj = mh. The dipole moment

coincident with the magnetic field can then be rewritten as in equation 4. Here the term

(eh/47rmc = pg) has been defined as the Bohr magneton. Therefore, the Zeeman energies

C eh
t = -2IT msc) (4)

for a magnetic dipole in a magnetic field are given in equation 5, where a quantum

electrodynamics correction needs to be included in lieu of the constant factor of 2. This

correction is deemed the g-factor (g = 2.0023).

E = -LH H = 23emsH = gJ3msH (5)

Quantum-Mechanical Treatment

The simplest, yet the most rigorous and instructive manner in which to introduce

the theory of Electron Spin Resonance (ESR) is to investigate the spin system of the

simplest chemical entity known; the interaction of an electron fermionn) with a proton

fermionn), routinely known as the ground state hydrogen atom ('H). In this case, the spin

of the electron ( S = /2 ) does not solely produce fine structure, but may also interact with







3

the spin of the nucleus ( I = /2 in this case) to give hyperfine structure and therefore

affords a rich example of this intricate interaction that is common to ESR. Furthermore,

the Hamiltonian for the spin degrees of freedom of a hydrogen atom is not complicated

by the anisotropies of the g-tensor and hyperfine interaction tensor (A) due to the lack of

spin-orbit coupling and the spherical nature of the electronic ground state. Thus, the

terms needed in the Hamiltonian are represented in equation 6.

S= geeH S + A ST I gnnHT I (6)

geeHT S T electron Zeeman term
AoS,, I hyperfine interaction term
gnf3nT = nuclear Zeeman term

When the z-axis is chosen to be the axis of space-quantization and coincident with

the magnetic field (H), the Hamiltonian transforms via equation 7. With the use of the

general raising and lowering operators (equations 8 and 9), equation 7 converts into

H = geeH Sz + A,(Sz Iz + Sy Iy + S" I x) gnnH Iz (7)

final functional form, which is represented in equation 10. The Hamiltonian that has been

+ = Jx + iy (8)

J- Jx iJy (9)

constructed will operate on the four independent basis functions I Ms, M ) for the


H- = geeH S+A. + o + + (0)

o- ato in z

hydrogen atom in the manner indicated in equation 11. The evaluation of the matrix









(Ms, M, 1- MS' M"1) (11)
elements have been worked out previously,' and the nonzero elements are contained
below in equations 12-16. The resultant 4 x 4 matrix, depicted in Figure 1.1, can be

(Ms S Ms) = Ms (12)

(M I1 Mi) = M, (13)

(Ms 1, MISIz M MI} = Mi[S(S + 1) Ms(Ms 1)]1/2 (14)

(Ms 1, M 1 SI+I Ms, MI) =
\n M-I(15)
[S(S + 1) -M,(Ms 1)]112 [Z(J + 1) M,(M, I)12 (15)

(Ms 1, M, T I SIJT Ms MM) =
^^ ,/7(16)
[S(S + 1) Ms(Ms 1)]112 [1(1 + 1) MI(Ms T 1)]112

solved as a secular determinant. To give a lucid representation of its solution, the matrix
has been block factored along the diagonal. The two (1 x 1) diagonal elements give the
following energies (equations 17 and 18), and the remaining (2 x 2) determinant can be
1 1 1
Ea(e)a(n) = + 9egeH + Ao I nPnH (17)
2 4 2

Ep(e)p(n) = 9eeH + A. + gnjnH (18)

expanded and solved to produce the energies shown in equations 19 and 20. Note that the
brackets that surround these particular eigenvalues (equations 19 and 20) indicate that
they are only true eigenvalues in the limit of high magnetic field. In reality, the two

states are linear combinations of the eigenstates [a(Sf,) and o] and their "purity" is











[ ot (e), a (n) > [ cc (e), 3 (n) >


< t (e), a (n)]





< ca (e), 13 (n)]





< 13 (e), ax (n)]





< p13 (e), 13 (n)]


1/2 g3H 4 1/4 A,
- 1/2 gnH


4.


1/2gp3H- 1/4 Ao
+ 1/2 gnf3,H




1/2 Ao


1/2 AO




- 1/2 gpH 1/4 AO
- 1/2 g3nH


- 1/2g311 1/4A,
+ 1/2 g,13nH


Figure 1.1. The spin Hamiltonian matrix for the Hydrogen atom.


[ 13 (e), 13 (n) >









1 2 2[(2 (2/2
E{a(e)(n)} = +A [(gePe + nP)2H2 + A (19

4 21 1
-- H Ao + A, )2 2 1i/2 (0
Ej3e)axn'} 4 A. 2 [(giePe + gnn)n + Ao (20)

contingent on the strength of the applied magnetic field. For instance, in a weak field the

states are heavily mixed and strong coupling occurs between the states, which gives rise

to four possible transitions.

The above solutions are referred to as the Breit-Rabi energies and are illustrated in

Figure 1.2 as a function of low, intermediate, and high magnetic fields. The upper panel
shows the low field region and the lower panel is at intermediate and high field. The

Breit-Rabi equation was originally utilized to explain the transitional behavior of

hydrogen atoms in a low-field molecular beam.2

The presence of off-diagonal terms in the matrix shown in Figure 1.1, necessarily
indicates that the original basis functions are not all eigenfunctions of the Hamiltonian. A

new basis set can obtained that are eigenfunctions of the Hamiltonian, which is based on

a coupled representation of the individual angular moment (see equation 21). This

F = S + I (21)
results in a total angular-momentum with a new set of quantum numbers, which can be

represented in ket notation as F, MF ). By inspection of the original matrix (see Figure

1.1), there are two functions that already exist as eigenfunctions by virtue of being

diagonal elements (I ae), a(e r) and I (e), P(n)), and they can be represented in the

coupled format as F, MF) = I 1,+l) and I l,-l). The remaining coupled








1 E/h (Ghz)
3 E/h (Ghz) ( -

0- 0 |--
c0r


: -7- 7
&: S \K --f/



ai
10
00
E l-\
cJQ' -> tJ\
s- q ~~~I') -
\ ffi/ 0
0 I/\









eigenfunctions F, MF) = 1,0) and 0,0) can be solved for by the expansion of the

(2 x2) matrix as mentioned previously and represented as linear combinations below

(equations 22 and 23).
The angle co in the equations below determines the weight of hyperfine and
Zeeman energies and can be extracted from the following relationships (equations 24 and

1,0) = Cos o(e)j', R,f)) + sin c) P(e, a ) (22)

0,0) = sinlco a(e), P(nl) + cos 01 P(e), a) (23)

25). At zero field, (H = 0, o t n / 4), the two states are equally mixed and the

sin 2a = (1 + 2)-1/2 (24)

cos 2(o = (1 + 42)-1/2 (25)

where 4 = (geJe + gn{3n)H / A,,

resultant linear combinations are given in equations 26 and 27. At the other extreme, as

H approaches infinity (H -- oo, co -+ 0) the two states collapse into separate and

distinct states (see equations 28 and 29).

1,0) = 1 2 a(eI (n)) + P(e), a(n))) (26)

I0,0) = 2 (I a(e), p(n) P(e),' Q)) (27)

I1,0) = a(e), P(n)) (28)

0,0) = P(e), a(X)) (29)







9

Transitions in an ESR experiment can be induced with a perturbingg" interaction

of electromagnetic radiation. A resonance condition is achieved when the frequency of

this incident radiation exactly matches the energy difference between an initial and final

state (equation 30). The intensity of this transition is the square of the matrix elements of

hv = Efinal Einitial (30)

the perturbing radiation between the initial and final states (equation 31). The operator in

|(initial 1Hil final)] (31)

equation 31, which represents the incident radiation, is defined formally (equation 32) as

the total dipole moment operator (_-T)dotted into the linearly polarized oscillating

magnetic field (H,). The above equation then transforms into final form as shown in

equation 33.

il = -_T H, (32)


I = (ge gnPn i) HI (33)

Low field transitions are quite different in terms of their selectivity and intensities

due to the coupling of states to give the effective quantum numbers: F, MF). In

essence there are four transitions with the selection rules of AMF = 1 by virtue of H1

being 1 to H ( when H // z and H, // x) where the transition operator becomes equation

34. A typical evaluation of a low field transition matrix element and corresponding

H1I = gAeHISx gnnHxi (34)

intensity is given in equations 35, 36, and 37 for the transition I 1,-1) 1,0). The










(P(e)' RnP() Ixk aIe), R(eI) = 1/2 (35)

(P(e), P(n) x P(e), a(n,)) = Y2 (36)

i1,-1) -+ I1,0) intensity a (gP, cos co gnP sin co)2 (37)

more common intermediate and high field transitions with H, // x and H, L H has a

transition operator as denoted in equation 38 with a general matrix element given in

equation 39. Thus, the selection rules for higher fields are in general AMs = 1 and AM,

= 0, and the intensity is the square of this element.

H1 = geHjS,, (38)

(Ms, M, H| Ms' M') = gTeHi(Ms 1xl Ms')(M I1 Mi') (39)

A typical experimental arrangement due to technological constraints is to utilize a

constant frequency source and to sweep a static field. A problem arises in this

experiment in that the magnetic field is different for subsequent transitions. An excellent

approximate solution3 to this situation is given in equation 40. With the values of g and

v, this equation can be solved for A.. It must be noted that ifA./g is > 0 then the M, =

-1/2 line occurs at fields higher than M, = +1/2, and the reverse is true if A./g < 0.

H = Ao 1
Ote 1 2
(2hv) 1/2 (40)

M { Mv2 + 2- Eh) ][A -(1 + J}













CHAPTER 2
THE NATURE OF A SPIN PROBE UNDER THE INFLUENCE OF SUPERCRITICAL
CARBON DIOXIDE (CO2)

General Description of Supercritical Fluids

The behavior of pure liquids and gases are in general fairly well characterized, but

the conceptual understanding of a supercritical fluid is somewhat esoteric. There has

been some evidence in the literature4'14 that supercritical fluids undergo "clustering" or

local density augmentation particularly near the critical point, which may explain some of

the unusual macroscopic behavior of these elusive fluids. A substance in the supercritical

state (especially approaching the critical point) seems to lose any homogenous identity by

undergoing time and spatial-dependent fluctuations in density. Extreme morphological

changes in the fluid with little or no change in the temperature or the pressure of the

system. (i.e.; the singular nature of the isothermal compressibility at this locale on the

phase diagram), would seem to indicate a struggle on the molecular level for the more

appealing "microscopic" phase of the moment.

These extreme molecular environments under the auspices of a single phase

affords these fluids unique properties that can be intermediate between a gas and a liquid.

This is evident by the fact that these fluids can possess the solvating power of a

condensed phase solvationn generally scales logarithmically with density), but on the

other hand may exhibit the mass transport properties diffusivityy and viscosity) of a gas.

11








12

This unique combination of solvent properties has led many authors"5 6 to describe the

nature of a supercritical solvent as tunablee." In this case, the tunabilityy" refers to the

thermodynamic timescale. But, what is truly happening on a faster timescale? To fully

realize the potential of supercritical fluids, a fundamental understanding at the molecular

level must be realized over a wide range of temperatures, pressures, and densities in and

near the supercritical region. Only after this has occurred will the potential of

supercritical fluids be revealed. A microscopic feel of these systems seems to be a

necessity. This is partly due to the extensive commercial interest in supercritical fluids as

a superlative alternative to traditional halogenated solvents because of their recyclability

and relatively benign activity towards the environment.

Introduction

The recent investigative fervor into the nature of supercritical fluids has created

some controversy1718 about the true behavior of these systems at or near supercritical

conditions. It is somewhat accepted that pure supercritical fluids possess some degree of

solvent-solvent clustering.4'-4 But, what is more speculative is the existence of solute-

solute association'9 when a dilute solute is introduced into a supercritical system. The

essence of the question is how does the supercritical solvent treat the impurity? Does the

supercritical system ignore the presence of the impurity and continue to self-cluster or

does the solvent fully solvate the solute as it clusters? Is there some degree of solute

association even in an extremely dilute situation where tiny time and spatial-dependent

reaction centers are created with the presence of a surrounding modulatory bath of

supercritical solvent structure? One can envision four or more possible scenarios:








13

1) solvent clustering with disregard to the solute impurity (without solute association) 2)

solvent density augmentation around individual solute molecules that tends to hinder or

enhance the transport of the solute (which may or may not be dependent on the solvent's

location on the phase diagram) 3) solute association with minor solvent clustering and 4)

solute association with solvent density augmentation around a solute cluster. These are

just some of the questions that we wish to address and to begin to answer in this

monograph.

The idea of critical clustering has been bantered about between authors who have

argued for and against its existence with seemingly varying degrees of conviction.

Randolph et al.17 initially reported evidence of critical clustering in and near supercritical

ethane based on enhanced spin exchange rate constants of a spin probe. On the other

hand, Batchelor"s has investigated the spin-rotation line broadening mechanism of very

dilute (-1 x 10"5 M) solutions of supercritical hexane and ethanol via a spin probe.

Batchelor"8 has argued that the spin-rotation mechanism is a more reliable indicator of

critical clustering rather than spin exchange on the basis that an enhanced spin exchange

can also be promoted via the onset of gas phase kinetics (and not necessarily critical

clustering).

More recently, Randolph et al.'9 have pointed out that without the presence of

solute-solute association, their results for rotational diffusion models do not produce

reasonable results. They have argued that previous investigators5'14 have ascribed

enhanced rotational diffusion times in various supercritical media to solvent-solute

density augmentation almost exclusively, but they (Randolph et. al.) stress that the








14

experimental results they have obtained do not make physical sense unless solute

association is occurring since the local density enhancement far exceeds liquid densities.

In this regard, the above authors have conceded previously the likely possibility that

solute-solute clustering is probably dependent on the Lennard-Jones interaction potentials

of the particular combination of solvent and solute.7

Thus, in order to begin to address the apparently rich phenomena outlined above

and to try to corroborate or dispute some of the claims of the previous investigators, we

have undertaken an investigation to explore some of the dynamical properties of a spin

probe (di-tertiary butyl nitroxide, DTBN) under the influence of supercritical carbon

dioxide (C02). Carbon dioxide has been chosen because of its amenable supercritical

conditions (T, = 31.0 C, P, = 1070.1 psi and p, = 0.467 g/mL )20 and its popularity as an

exemplary alternative solvent.

Experimental

A high-pressure ESR cell fabricated (see Figure 2.1) from 6 mm O.D. and 2 mm

I.D. quartz capillary tubing traversed a transverse electric (TE102) ESR cavity and was

mounted horizontally (The ESR cavity was mounted in this fashion to minimize the

gravitational effect on the concentration near the critical point,2' and all of the remaining

components of the system are coplanar to within approximately 1") between the poles of

an electromagnet. The ESR cell was connected to ancillary components (see Figure 2.2)

of the system via 1/16" stainless steel tubing. This entire system was enclosed by a

carefully constructed insulated foamboard enclosure and was subsequently sealed with a

commercial foam sealant to create a semi-permeable insulating structure that would fully














7 -~


III


TFL-


1-Iart-


Figure 2.1. The high pressure ESR cell.


. .bias


V ^-B












A:
B:
C:
D:
E:
F:
G:












To Cylinder


<- Insulated Structure









To Vacuum


Figure 2.2. The block diagram of the high pressure ESR apparatus.


I-I








17

encompass all of the above mentioned components. The temperature of the subsequent

air bath was regulated and controlled through the operation of two power resistors

(nominally 110 W) that were suspended within the insulated volume and three 6"

circulatory fans that were run at a high rate for thorough mixing. The temperature was

monitored by two calibrated resistance temperature devices (Omega, model

1PT100K2010) that were placed within the enclosed environment. The air bath was

thermostatted to within 0.2 C. The pressure of the apparatus could be varied by a

piston (High Pressure Equipment, standard model 62-6-10) and monitored with a strain

gauge pressure transducer (Omega, model PX612 5KGV) to within 2 psi.

To obtain supercritical conditions of carbon dioxide the apparatus was first

evacuated and flushed several times to ensure that contamination did not occur. The

vessel was then charged with fresh CO2 (Scott Specialty Gases, Instrument Grade

99.99%) up to the maximum pressure of the CO2 cylinder and was further pressurized

with the use of liquid nitrogen to facilitate the condensation of CO,. The temperature

was then brought up to the desired supercritical isotherm and a specific pressure was

obtained.

At this point, a high pressure circulating pump (Micropump, model 1805T-415A)

was initiated to begin the circulation of the CO2 through the high pressure system. An

equilibration period of at least two hours ensured that the fluid had reached thermal

equilibrium. After this allotted period, di-tertiary butyl nitroxide (DTBN, Aldrich,

90%tech. grade) was introduced into the volume via a high pressure injection valve

(Valco, model C2-2306) equipped with a 20 p.L injection loop. The circulating pump










ensured mixing of the DTBN and carbon dioxide. The pressure and temperature were

then noted as the starting conditions, and the concentration of CO, was calculated by the

appropriate P,Vm,T measurements (based on a total volume of 140 1 mL ) by Wagner

et.al.20 The accurate determination of the CO2 concentration was needed to calculate the

mole fraction of DTBN relative to CO2. The weight fractions of DTBN to CO2 were well

within the range of solubility.22

After the equilibration period, ESR spectra were recorded with the following

experimental arrangement: an X-band frequency generator (Varian El 02 Microwave

Bridge) was coupled to a horizontally mounted TE 102 cavity and placed 90 relative to a

permanent static field (Varian V-3601 Electromagnet). The field was modulated at 100

kHz with a modulation amplitude of 1 G. The center field was set at 3225 G and swept

over a 50 G range. After a spectrum was recorded at a particular pressure along an

isotherm, the mixture of carbon dioxide and DTBN was slowly leaked via a leak valve

from the system to obtain a new pressure. In this way, a constant mole fraction of CO2

and DTBN was maintained in order to monitor the behavior of supercritical CO2 along an

isotherm over a wide range of pressures (and thus densities) to see the response of the

fluid with a constant number of probe molecules at each pressure. This procedure would

give a similar thermal profile for each pressure and density.

Results and Discussion

The two dominant line broadening mechanisms that contribute to the overall

linewidth in these systems will be spin-rotation at dilute concentrations and spin










exchange at higher concentrations. It has been pointed out'8 that spin-rotation might be a

better indicator of critical clustering rather than spin exchange because enhanced spin

exchange near the critical density might just be an indication of the onset of gas phase

behavior rather than a sign of clustering. A deconvolution of these two line-broadening

mechanisms is difficult in concentrated samples because of the overwhelming

contribution from spin-exchange, which tends to mask the residual linewidth (and thus

the contribution) from the spin-rotation. Therefore, solute-solute and/or solute-solvent

clustering might best be explored with the observation of spin exchange (Vex.) and

correlation time (T,) simultaneously (with the treatment of spin-rotation as a residual

Gaussian linewidth contribution). In this manner, clustering pertinent to the spin probe

will be shown more conclusively than by study of either independently. Correlation

times essentially reflect the extent of time-averaging of the anisotropies caused by the

modulation activity of the environment surrounding the spin active species. Therefore,

correlation time will manifest itself in a completely different manner (essentially an m,

dependence in the ESR spectra) in relation to spin exchange (usually an equal broadening

of all lines considered).

To try and circumvent the above mentioned problem, a "modest" amount (~ 1E-5

mole fraction) of spin label (DTBN) was introduced into supercritical carbon dioxide

(COD) to try and account for both the spin exchange (v,.) and correlation time (Ta). It has

been reasoned that the dominant mechanism at higher pressures should be mostly spin

exchange, but as the pressure is decreased towards the critical pressure, spin-rotation








20

might have a more significant contribution. Specifically, spin-rotation should be less

obvious at higher pressures, and more pronounced at lower pressures as there will be a

marked decrease in viscosity (ri) as an isotherm is traversed (spin-rotation has a Th/

dependence). Therefore, if normal behavior prevails spin-rotation will contribute to an

overall larger linewidth as pressure decreases. Therefore, the spin exchange frequency

that will be modeled might be slightly higher (due to the increased contribution of spin-

rotation at lower pressures) than the absolute frequency, but nonetheless the rate constant

of spin exchange (k,) versus correlation time (Tr) should still show the trend sought.

Otherwise, increased spin exchange would not necessarily point to solute association,

because it might simply be explained by the onset of gas phase behavior and an increase

in transport phenomena. Therefore, the prudent representation of the data would be the 3-

D plot of spin exchange (v,) and correlation time (to) versus pressure (or density).

The general strategy was to prepare a supercritical bath at selected pressures (and

thus densities) along an isotherm and to extract the motional behavior (i.e.; v,, T, and

residual linewidth (lw,,.)) of the spin probe from ESR spectra. This would be performed

over several temperatures and mole fractions to determine their dependence or lack

thereof on the dynamical properties.

The ESR spectra were simulated with the use of modified version of Freed's'

simulation program that have been tailored specifically for nitroxide spin labels. Freed's

program explicitly accounts for spin exchange and correlation time and all other

relaxation processes can be taken into account by a residual linewidth. The simulation of









these spectra were performed by inputting the g and A tensors of DTBN extracted from

the powder spectra of Griffiths and Libertini.24 The spin exchange rate (v,.), diffusional

rate coefficients (dvy, d,), and the residual linewidth (lw,,) were varied to optimize the fit

of the experimental spectra. The spin exchange rate and diffusional rate coefficients were

used to calculate the respective rate constants of spin exchange (ke) and rotational

correlation times (To) appropriate for each fit (see equations 41-43).


Vex = ke [DTBN]2 (41)


(dxydzz) 1 = drot (42)


1
TC 6drot (43)



Figures 2.3-2.7 show the series of digitized ESR spectra with their respective

simulations for a series of pressures along the 34 C isotherm at X = 6.1E-5 (where X is

defined as mole fraction). Table 2.1 shows a summary of the pertinent thermodynamic

and motional data for these spectra (Figures 2.3-2.7). Figures 2.8-2.11 show a series of

spectra and simulations again at 34 C isotherm, but at X = 5.4E-5. Table 2.2 shows a

summary of these data. Figures 2.12-2.15 show spectra and simulations at 40 C (X = 5.4

E-5) and Table 2.3 shows a summary of these spectra. Finally, Figures 2.16-2.18 show

50 C spectra (X = 6.4 E-5) and Table 2.4 shows a compilation of these data.








22

Figures 2.19-2.24 illustrate the nature of the correlation time and spin exchange

behavior with respect to density, temperature, and concentration. At all three

temperatures there is a general increase in correlation time (T) as the reduced density is

traversed from high density to low density, except for the 34C isotherm at X = 5.4E-5

(compared to X = 6.1E-5). Here there is a general increase in the correlation time until

the reduced density approaches approximately 0.68.

The rate constants (ke) also increase in general as reduced density is traversed

(except for the above mentioned example) for the 34 C and 40 C isotherms. The 50 C

isotherm shows a general increase then a small decrease in rate constants that mirror a

subtle increase in correlation time.

Figure 2.24 shows the temperature dependence of the correlation times at a

constant mole fraction of 5.4 E-5. This comparison suggests that the interaction of the

solvent becomes more frequent at the lower temperature (34 C) at an inflection point of

reduced density at approximately 0.68 and nearly equals the correlation time found at the

higher temperature (40 C) at this density. Overall, the correlation time is higher for the

lower temperature up until this point.

The temperature dependence of the rate constants (Figure 2.23) at the same mole

fraction of 5.4 E-5 indicates that the spin exchange rate is lower in general for the lower

temperature until this reduced density is reached (0.68). At this point, a precipitous

increase in the rate constant is seen for the 34 C isotherm that slightly exceeds the rate

constant observed at the 40 C isotherm.










Conclusion

It appears that the rate constants (k1) for spin exchange generally increase as the

reduced density decreases at the lower temperatures (34 C and 40 C) nearer the critical

temperature (Tr = 32.1 C). A precipitous increase in k, is witnessed at the 34 C

isotherm and mole fraction of X = 6.1 E-5 at a reduced density of approximately 0.68. A

concomitant increase in correlation time (,r) is generally prevalent at these temperatures

also. This behavior indicates a clustering phenomenon (either solvent-solute or solute-

solute), otherwise the correlation time should decrease as the rate constant increases if gas

phase behavior is truly the cause of an increased rate constant. The exception to this

behavior is the lower mole fraction (Q = 5.4 E-5) at 34 C. At a reduced density 0.68,

there is a distinct decline in T- with an increase in k. This would indicate an onset of gas

phase behavior instead of clustering and points to solute-solute clustering as a possible

cause of enhanced rate constants in the higher mole fraction (Q = 6.1 E-5) experiment at

34 C.

The behavior at 50 C indicates that there is a marked increase in rate constants

and correlation times at a higher reduced density (PR 1.0) and then a leveling off at

lower reduced density. This would indicate that clustering behavior might occur at higher

reduced densities when the temperature is further removed from the critical temperature.

The related errors in the spin exchange rate constants (k1) and correlation times

(-;) that are inherent in the simulations can be accounted for with a generous range of

0.5 Lmol's'sxl0'13 for ke and 5 ps for T. respectively.









0.02

o1w%
C
0.01.


I 0.00- J


S-0.01
Q
-0.02-


-0.03 ,,,
-30 -20 -10 0
Magnetic Field (G)


Figure 2.3. The ESR spectrum of DTBN in supercritical CO2 at T = 34 C and P = 1275 psi.


0 20 30
I020 30









0.02-


= 0.01


S0.00


S-0.01


-0.02


-0.03 ,
-30 -20 -10 0 10 20
Magnetic Field (G)


Figure 2.4. The ESR spectrum of DTBN in supercritical CO2 at T = 34 C and P = 1167 psi.












C


e
C


0


-30
-30


S I 1 -
-20 -10


0 10
Magnetic Field (G)


Figure 2.5. The ESR spectrum of DTBN in supercritical CO2 at T = 34 C and P = 1090 psi.


0.02-


0.01-


0.00-


-0.01 2


-0.02 v V
-0.03


20


' I0
30


E

















.Q
0


0.02

0.01

0.00

-0.01

-0.02

-0.031


-30 -20 -10 0 V
Magnetic Field (G)


Figure 2.6. The ESR spectrum of DTBN in supercritical CO, at T = 34 C and P = 1070 psi.












I


C
=.2


.0
0
CO
<


0.02


0.01


0.00


-0.01


-0.02


-0.03--
-30


Magnetic Field (G)


Figure 2.7. The ESR spectrum of DTBN in supercritical CO2 at T = 34 C and P = 1050 psi.










Table 2.1. Summary of thermodynamic and dynamic properties of C02/DTBN along the T = 34 C isotherm
with a X = 6.1 E-5 mole fraction.


Pressure (psi)


1275

1167

1090

1070

1052


p (g/mL)


0.672

0.576

0.294

0.271

0.252


PR [ ]rDTRN (M)


1.43

1.23

0.629

0.580

0.539


9.21 E-4

7.97 E-4

4.09 E-4

3.76 E-4

3.50 E-4


ve. (s-") k, (Lmol-'s-) x 1-D


2.0 E7

2.0 E7

2.3 E7

2.4 E7

2.5 E7


T, (ps) lwr,,s (G)


2.3

3.1

13.7

16.9

20.4









0.04


0.02


e 0.00- ___


-0.02-


-0.04-


-0.06- 10 2 .
-30 -20 -10 0 10 20 30
Magnetic Field (G)

Figure 2.8. The ESR spectrum of DTBN in supercritical CO2 at T = 34 C and P = 1545 psi.













C




0
el




o
CO)
.Q


0.04-


0.02-


0.00-


-0.02


-0.04


-0.06-

-30


I
-20


-10


Magnetic Field (G)


I
10


Figure 2.9. The ESR spectrum of DTBN in supercritical CO2 at T = 34 C and P = 1350 psi.


20
2O


30


8








0.06-

0.04

0.02

0.00

-0.02

-0.04

-0.06

-0.08--


i i i ^i^r
-30 -20 -10 0 10
Magnetic Field (G)

Figure 2.10. The ESR spectrum of DTBN in supercritical CO2 at T =34 C and P = 1100 psi.


EL
CO


0

0
&'
-a


20


I
3O







0.06-

0.04

0.02

0.00.

-0.02

-0.04

-0.06 -
-30 -20


I 01
Magnetic Field (G)


10 20 30


Figure 2.11. The ESR spectrum of DTBN in supercritical CO2 at T = 34 C and P = 1080 psi.



I=
C




0
1
2








Table 2.2 Summary of thermodynamic and dynamic properties of C02/DTBN along the T = 34 C isotherm
with a X = 5.4E-5 mole fraction.


Pressure (psi) p (g/mL) PR [I DTN (M) v, (s-') k, (Lmol-'s-') x 10-13 T, (ps) lwres (G)


1545 0.746 1.59 9.21 E-4 1.0 E7 1.1 23 0.8

1350 0.700 1.49 8.59 E-4 1.1 E7 1.4 27 0.7

1100 0.311 0.665 3.82 E-4 1.0 E7 6.8 37 0.65

1080 0.282 0.603 3.45 E-4 2.1 E7 17.6 25 0.35







0.02


| 0.01


0.00-


S-0.01


-0.02


-0.03 ,- -,---,
-30 -20 -10 0 10 20 30
Magnetic Field (G)

Figure 2.12. The ESR spectrum of DTBN in supercritical CO2 at T = 40 C and P = 1895 psi.








0.02 /

C 0.01

0.00
|= o-oo- t r/


S-0-01-
ri
< -0.02

-0.03- -. .
-30 -20 -10 0 10
Magnetic Field (G)
Figure 2.13. The ESR spectrum of DTBN in supercritical CO2 at T = 40 C and P = 1547 psi.








0.02


C 0.01-

LO 0.00 -


~. -0.01.


-0.02


-0.03- |,,,,,
-30 -20 -10 0 10 20 30
Magnetic Field (G)


Figure 2.14. The ESR spectrum of DTBN in supercritical CO2 at T = 40 C and P = 1355 psi.











3 0.02-


i 0.00-


". -0.02


-0.04-
--- ---- -- --T-- --1-- -- F
-30 -20 -10 0 10
Magnetic Field (G)

figure 2.15. The ESR spectrum of DTBN in supercritical CO2 at T = 40 C and P = 1160 psi.








Table 2.3 Summary of thermodynamic and dynamic properties of C02/DTBN along the T = 40 C isotherm
with a = 5.4E-5 mole fraction.


Pressure (psi) p (g/mL) PR [I ]DTBN(M) v (s-1) 1k (Lmol-'s') x 10-3 T, (ps) lw. (G)


1895 0.745 1.59 9.21 E-4 3.5 E7 4.1 16 0.5

1547 0.669 1.43 8.22 E-4 3.0 E7 3.4 18 0.5

1355 0.561 1.20 6.89 E-4 3.0 E7 6.3 23 0.6

1160 0.282 0.603 3.47 E-4 2.0 E7 16.6 27 0.55








0.04-



60ft 0.02-
:2


0.00



; -0.02



-0.04- ,,,,,
-30 -20 -10 0 10 20
Magnetic Field (G)

Figure 2.16. The ESR spectrum of DTBN in supercritical CO2 at T = 50 C and P = 1875 psi.







0.03-

0.02
C
,' 0.01
3
e o.oo-
r .0.00-

0
s -0.01-

< 4-.02

-0.03

-30 -20 -10 0 10 20 30
Magnetic Field (G)


Figure 2.17. The ESR spectrum of DTBN in supercritical CO2 at T = 50 C and P = 1555 psi.









ot o / Ii1
0.02

C
.e 0.00.
S

2 -0.02


-0.04- .-,-
-30 -20 -10 0 10 20 30
Magnetic Field (G)


Figure 2.18. The ESR spectrum of DTBN in supercritical CO2 at T = 50 C and P = 1343 psi.








Table 2.4 Summary of thermodynamic and dynamic properties of C02/DTBN along the T = 50 C isotherm
with a X = 6.4E-5 mole fraction.


Pressure (psi) p (g/mL) PR [ JDTBN(M) v.(s-') ke (Lmol-'s-') x 10-"3 T (ps) lWs (G)


1875 0.633 1.35 9.21 E-4 3.2 E7 3.7 8 0.5

1555 0.471 1.01 6.85 E-4 2.2 E7 4.6 16 0.6

1343 0.307 0.657 4.46 E-4 9.0 E6 4.5 18 0.6

















-- spin exchange rate constants (ke)
--- correlation time (Tc)


15

S
B

.-10.





5.





0.


Preduced


Figure 2.19. The dynamic properties at T = 34 C and X = 6.1 E-5.


254.


204.


I I I I I















20 40


18 --- spin exchange rate constants (ke)
correlation time (Tc)

16 35


14-


S12 30

10..
-" 10 \-
S
8 8 25


6.


4-- 20


2


0 I III I I 15
0.4 0.6 0.8 1 1.2 1.4 16

Produced

Figure 2.20. The dynamic properties at T = 34 TC and X = 5.4 E-5.















I -- spin exchange rate constants (ke)
-- correlation time (Tc)


16


14


124


I I I


Reduced


Figure 2.21. The dynamic properties at T = 40 C and X = 5.4 E-5.


20r















I --- spin exchange rate constants (ke)
-U- correlation time (Tc)


4.74.


4.1 4.


3.94.


Figure 2.22. The dynamic properties at T = 50 C and X = 6.4 E-9'.'


4.94


3.5 1---)
0.45 0.65 085 105 1.25 145













18.


16

--T=34C
14 -i-T=40C


,3 121


10,



8.


6


4.


2

0 I I I II

0.4 0.6 0.8 1 1.2 1.4 1.6

Produced
Figure 2.23. The temperature dependence of the rate constants (ks) at = 5.4 E-5.
















--T = 34 C
---T=40 C


0.4 0.6 0.8 1 1.2 1.4 1.6

Produced
Figure 2.24. The temperature dependence of the correlation time (,) at X = 5.4 E-5.













CHAPTER 3
VIBRATIONAL SPECTROSCOPY THEORY

Classical Description

The origin of vibrational activity stems from the seemingly simple interaction of

two mass points that are physically connected through space and are originally at an

equilibrium position (see Figure 3.1). The impending motion (in the absence of torque or

any rotational motion) can be described by classical harmonic motion, which is defined as

the movement that results when a force acting on a body is proportional to the

displacement of the body from an equilibrium position2 5 Implementing Newton's First

Law to describe this motion (see equation 44) results in a homogenous 2nd order

differential equation. The general solution to type of differential problem is given in

a2
ax
= ma = m -kx (44)
at2

equation 45. If this eigenfunction is differentiated twice, the result is given in equation

x = A cos (Cot + 6) (45)

46. The angular frequency associated with the linear motion of this system can be related

m(-o 2 [A cos (cot + 6) ] ) (46)

simply to the ratio of the force constant (i.e.; the scalar response to the force acting on the

body) to the mass (shown in equation 47). The one-dimensional energy of a two-body

50




















Xl


x

center of mass


Figure 3.1. The two-body oscillator.


X2










v = ( (47)
27t n

oscillator can be represented as shown in equation 48.

1 1 2 (48)
T = mv12 + v + -cx (4k)
2 n r2 2

v1 linear velocity of m,

'72 a linear velocity of m,

1 2 axV
kx = classical potential integrated from F = --
2 ax

By using the center-of-mass relationship (i.e.; m,x, = -mx,), differentiating with

respect to time, and utilizing moment in lieu of mass and velocity (equations 49 and 50),

dxl dx2
ml dt -m2 dt (49)

mlnv1 = -m2V2 = Pi = P2 (50)

the energy can be represented in terms of moment (equation 51). The energy can be

pI2 p22 I
T + 2 m+22 + kx2 (51)
2m, 2Mn2 2

eventually reduced to a representation (equations 52 54) involving momentum (p) and
1= F2P2 + 1P 1 2
I [ + -kx2 (52)
2 MI m2 2

m1in2
9= MI (53)
MI + Mn 2

reduced mass (gi), which is commonly known as the classical Hamiltonian.










p2 1
T = + kx (54)
2ut 2

Quantum-Mechanical Treatment

The classical Hamiltonian for a two-body interaction can be transformed into the

quantum-mechanical operator via equation 55. The Hamiltonian in essence contains all


-- V2 + V(r)1 = Ey (55)
2u

of the energetic information of the system, but the center-of-mass contribution to the

kinetic energy operator in this instance has been excluded in this equation because it

represents only a shift in the total energy of the system. When the cartesian coordinate

system (x. y, z) is reconfigured into spherical polar coordinates (r, 0, ( ), the Hamiltonian

converts to equation 56, where .2 is the square of the angular momentum. The solution

of this equation involves the selection of an appropriate wavefunction that is separated

h2 [ 1 a 2 w 1 (6
2 r2 r + 2 J + V(r_ = E_ (56)
2 r ar arJ 2r2



into a radial and angular dependence (see equation 57, where Yj.M is the spherical

harmonic).

1 = R(r)Yj, (M,) (57)

The one-dimensional radial Schrodinger equation is the result of this substitution

(equation 58). An effective potential can be written to encompass the potential due to the









intemuclear separation (V(r)) and a "centrifugal" potential26 of the spherical harmonic
ft2 d 2dR? ^2J(J + 1) } _
r2- -_ d 2 + h2jj+ + V(r) JR = ER (58)
2g dr dr 2Iltr2

Veff= V ( + Vce, (59)

(equation 59). To simplify equation 58, the substitution of S(r) = R(r)r gives equation 60.

h2 d2S jV + 1)
-+,2 2 + V(r S = ES (60)
2pr2 dr2 2+r2

A rigorous solution to this equation for the corresponding energy levels and

wavefunctions requires only a functional form of the potential (V(r)). Generally speaking

the potential V(r) encompasses the electronic energy Eel (r) (obtained from the solution of

the electronic Schrodinger equation) and the nuclear repulsion term (Vx) (see equation

61). The parametric dependence of the electronic dependence Ee, on r (and thus the lack

V (r) = Ee, (r) + VrN (61)

of an analytical form for Eei and VNN) has led to the empirical development of V(r). A

popular potential (Dunham potential26) is a Taylor series expansion about r, (equation

62). Typically V(r,) is set equal to zero (chosen arbitrarily) and therefore the first
12
(r) = V(r) + r r) + (r re)+... (62)
dr re 2 dr2 r

derivative is zero (equation 63). If the first term is the only one retained (equation 64) in

the expression and J is set equal to zero, the harmonic oscillator solutions are obtained

(equation 65), where H(ca'ax) are the Hemite polynomials. The related eigenvalues are










dV = 0 (63)
drer

given in equation 66 for the non-rotating harmonic oscillator.

1
V(r) = k(r re)2 (64)
2

d2V
k d-r-r
dr2
re

ar 2
S = NvHv(a)l1/2xe 2 (65)

x = r -re
pgO)

S2v --



E(v) = hv(v + 1 / 2) (66)

i (k) 12
27c \Lp

Still another possible potential is the Morse potential (equation 67), which

approaches a dissociation limit [V(r) = D] as r goes to infinity. Furthermore, the Morse

potential can be solved analytically to give the following eigenvalues with harmonic and

V(r) = D(l e- (r-re)2 (67)

anharmonic terms for the harmonic oscillator (equation 68).

1 12 (68)
G(v) = CDe(V + -) (OeXe(V + ) (68)
2 2







56

The prediction of allowed vibrational transitions involves the assessment of the

dipole moment integral (equation 69) in which single primes denote upper levels and

double primes denote lower states. If the dipole moment (p(r)) is represented as a Taylor

series expansion (equation 70), the evaluation of this integral is given in equation 71. It

is immediately evident that the first term on the right is equal to zero or 1, because the

MV' v" = J (p'vib (rkP"vib dr (69)

A 1 d2^
d e + -I (r re) + P (r re)2+. (70)
= Le + rrr e +2dr2 2 "
e e


Mv' v" = efJ (Pvib "vib dr + d J 'vib (r re)P"vib dr+.. (71)
dre

vibrational wavefunctions are orthogonal. Of the remaining terms, the second term

(which is the change in the dipole with respect to change in position evaluated at the

equilibrium position (re)) contributes the most to the intensity of the transition (see

equation 72).

I o Iv'v"2 d^t 2
I c Mv' V"|2 oc d, (72)
dr

The remaining component that needs to be evaluated for a vibrational transition is

the integral in equation 73. With the utilization of the harmonic oscillator wavefunctions

and the recursion formula between Hermite polynomials (equation 74), the resultant

integral is given in equation 75. The evaluation of this integral leads to the familiar


qI Pib (r rek"vib dr


(73)







57
2xHn(x) = Hn+ I (x) + 2nHn_- 1(x) (74)
vibrational selection rules of Av = (fundamental bands only), because the Kronecker 6
(in equation 75) is v' = v +1 or v 1.

(V' H V)= (hj )I/2[v + vv+ + vv] (75)













CHAPTER 4
INFRARED SPECTRA OF Nb2C, Nb3C, and NbO, MOLECULES ISOLATED IN
RARE GAS MATRICES

Introduction

The rudimentary architecture of "met-cars"27 is the metal carbide diatom. Within

this frame of reference, it is necessary to elucidate the basic interactions between the

metal and carbon as a duet with and without the perturbing interactions of a solvent

structure. With this in mind, the gas phase ground electronic state of Nb12C has recently

been determined to be 2A3/12 by Simard et. al.,28 confirming the original prediction of

Weltner and Hamrick29 based on electron spin resonance (ESR) studies of this radical in

solid rare-gas matrices. In this specific case, it would have proven difficult to detect (via

ESR) such a species in a rare-gas matrix without the "quenching" of the orbital angular

momentum. A diatomic radical with an orbitally degenerate ground state (i.e.. [I, A, etc.)

is typically rendered undetectable in the ESR due to the subsequent diffuse nature of the

signal. But, in some instances angular momentum can be "quenched" by an

orthorhombic crystal field' which causes its properties in the matrix to approach that of a

I molecule.

It is generally the case that rare-gas matrices at cryogenic temperatures provide

the "mildest" of perturbations on the trapped guests,30'3' indicated by shifts from their gas

phase vibrational frequencies and electronic levels being only a few percent.30'

58








59

However, if the molecule is highly ionic (as, for example, LiF) the molecule-matrix

interaction can be large and the shifts in the solid state considerably larger. If this

interaction is substantial and the molecule is sitting in an asymmetrical site in the matrix,

then large anisotropic crystal field effects can also occur which effectively remove the

axially-symmetric character of the electronic wavefunction, as referred to above. And,

generally speaking the chemistry in the gas phase versus the condensed or solid phase can

be radically different as is evident in gas versus condensed phase acidities of some

mineral acids (e.g., HCQ). The entrapment of these molecules in solid rare-gas matrices is

integral to the complete understanding of the entire continuum of the chemistry between

the isolated gas phase intrinsic properties and the intimate guest/host interactions of the

solid and condensed phases.

The ground state of NbC has been calculated to have an exceptionally large dipole

moment (p. = 6.06 D)28 so that even without the quenching of the orbital angular

momentum one can expect large matrix shifts for corresponding optical spectra when

investigated in a matrix. Also, through spin-orbit effects there can be mixing of the lower

states so that forbidden electronic transitions may be observable. With this in mind, we

now wish to report the effects of the ground state vibrational frequencies (AG,,") of

Nb2C, NbVC, and NbO2 when trapped in solid neon, argon, and krypton.

Exprimental

The experimental setup has described in detail previously.32 In summary,

mixtures of niobium and carbon (slightly rich in the metal) were pressed into pellet form








60

and vaporized with a highly focused Nd:YAG laser (Spectra Physics DCR-1 1) operating

at 532 nm. The metal and carbon plume was co-condensed in rare gas matrices at a rate

of approximately 10 mmol/hr over periods of 1 to 1.5 hour onto a gold-plated copper

surface at 4 K. The IR spectra were measured via a vacuum FTIR spectrometer (Bruker

IFS-1 13V) equipped with a liquid-nitrogen-cooled MCT detector (400-4800 cm') used in

conjunction with a KBr beam splitter. The spectra were taken with a resolution of 2 cm'

(or in some cases 1 cm') and a scan number typically of 200.

The niobium powder was purchased from Electronic Space Products International

(99.9% purity). Carbon-12 was obtained as a spectroscopic grade electrode in graphite

form and ground into a fine powder. Amorphous carbon-13 (99% purity) was purchased

from Isotech and outgassed at 1400 C for 1 hour before use. The matrix gases were

obtained from the following vendors; neon (Matheson, 99.9995% purity); argon (Airco

99.999% purity); and krypton (Praxair, 99.9985% purity).

Results and Discussion

Kr

The most conclusive evidence for the formation of Nb'2C and Nb3C was in a

krypton matrix shown in Figure 4.1 (and Table 4.1). Traces A and B show the absorption

spectra in the stretching regions for the fundamental bands of Nb'2C (B) and Nb3C (A)

after annealing the spectra to 45 K and quenching to the original temperature of the

deposition (4 K). By observing the similarities and differences between the two spectra,

one can discern that the bands to the blue of the sharp band at 941.0 cm' in trace B are












S*NblC andNb'C


* A


1100


950 900 850 800 750 700 65(


Wavenumbers (cm- 1)


Figure 4.1. The IR spectra of Nb'2C and NbVC in krypton.


1150


. . . I I I I








62

common to spectra A. Therefore, it is quite reasonable to remark that these absorptions

are not due to any NbC containing products. The clear differences between the two

spectra are: 1) the absorption at 941.0 cm' in the lower trace and 2) the absorption at

908.5 cm' in the upper panel, which corresponds very reasonably with the isotopic shifts

that can be calculated by the usual harmonic relationship for diatomics represented in

equation 76. Both of these absorptions are quite strong in their respective spectra and are


V13 = V12 12 (76)


absent in comparison with the isotopic counterpart. Therefore, these two absorptions

have been confidently assigned to Nb'2C and Nb3C respectively.

The strong peak at approximately 949 cm' has not been assigned, but it appears in

traces A and B, therefore it is assumed that it is not a Nb()C( ,) species. It has also been

assumed that the absorptions (the weak doublet at 970.3 and 967.6 cm' ) that are at higher

wavenumbers relative to the NbC species are due to the stretching frequency (ies) of NbO

and/or NbO2. These transient species are prevalent in this type of experiment as has been

remarked by Simard et. al.28 and the personal experience of the investigators.

Unfortunately, the characterization of these species in a krypton matrix has not been

made to date, so it cannot be concluded for certain that these are NbO() impurities. The

belief that this absorption is indeed due to NbO is based on the fact that these peaks are at

slightly lower frequencies than those assigned to NbO in neon and argon matrices,33'37

which will be discussed further below. The weak absorption at 1024.5 cm' is again not

believed to be a Nb(U)C( ) because it appears periodically in both Nb'2C and Nb3C spectra.












Figure 4.2 displays the spectra from experiments carried out in a argon matrix.

Traces A and B show the results of the Nb3C and Nb2C experiments respectively. The

stretching frequency due to Nb'2C is assigned to the band at 952.2 cm'. The absence of

this peak in the Nb3C spectra (trace A) gives further proof of this assignment. Likewise,

the absorption at 917.7 cm' that is clearly evident in the Nb3C traces is amiss in the

Nb'2C spectra.

Again, there are residual peaks blue-shifted relative to the above assignments. It

can be said with certainty in this case that some of these are due to the fundamental

modes of NbO. Green, Korfmacher, and Gruen33 have previously assigned these peaks

(971 cm', 968 cm', and 964 cm"') to NbO in an argon matrix. The three peaks have been

attributed to NbO in different sites in the argon lattice. Note there is an extra peak at

974.6 cm' that appears in the matrix containing Nb "C.

In the Nb2C trace it is evident that there are some transitions besides those

attributed to Nb2C and NbO that do not appear in the top trace. The doublet that appears

at 946.2 and 941.4 cm' could possibly be Nb(n)C, but the corresponding "3C substituted

frequencies should be present and are apparently absent. Still, another possibility is that

the Nb2C peaks are from molecules in different sites in the matrix. The relatively strong

peak to the red of the assigned Nb3C peak (917.7 cm') at 911.5 cm'" (which may also be

the peak in the Nb2C trace at 913.3 cm') cannot be definitely assigned. The two sets of

triplets (913.1,907.2, 901.1 cm-' and 873.8, 867.8, 859.6 cm-') that appear in the Nb2C









NbC andNb'C
NbO and Nb02

A

C.
x*
x





S I* S S S S SS
,u^\i^^JA ll
a "it
IILA

i

B
C:S xx






1200 1150 11 00 1050 1000 950 900 850 800 750 700 650

Wavenumbers (cm-7)
Figure 4.2. The IR spectra of Nb2C and Nb3C in argon.










spectrum, but do not appear in this specific Nb3C trace (but have been observed in other

spectra of Nb3C) have been assigned to NbO2, since it has been reported that in certain

Knudsen cell/mass spectrometric experiments34 that equal amounts of NbO and NbO2 are

observed to form on vaporization of Nb205. Further evidence for this conjecture is that

the triplet pattern (presumably three sites) for both of these transitions mirror the behavior

of NbO. If this is true, it is the first reported evidence of NbO, in a matrix experiment.

This will be discussed further. Finally, the peak at 1030 cm' is a transient peak that

appears in both experiments although it does not appear in trace B for this particular

experiment.

It is interesting to note that the gas phase vibrational frequency for NbO (AG|^")

has been determined to be 981.37 cm' by Gatterer, Junkes, Salpeter, and Rosen.35-36 This

value is extremely close to the gas phase ground state vibrational frequency of Nb2C

(AG11/2" = 980 15 cm-') given by Simard et. al.28 With all else considered, the position of

the vibrational frequency in various matrices should be a measure of the relative strengths

of interaction between NbC (and NbO) and the matrix.

Ne

The results in neon (Figure 4.3) are the least revealing. In trace B, the weak

absorption at 983.0 cm' is tentatively assigned to Nb'2C. In the upper panel, low

frequency shoulder of the doublet at 952.8 cm' and 949.7 cm' is assigned to the Nb3C

stretch, based strictly on the calculated isotopic shift from the band at 983.0 cm"'

Furthermore, the gas phase value given by Simard et. al.28 is very close to the assignment












*Nbl2C andNb13C
XNbO


B






H e S S g I | S S S I a i ||


1050 1000


950 900


850 800 750


70 650


Wavenumbers (cm-I)


Figure 4.3. The IR spectra of Nb'2C and Nbl'C in neon.


120


U


1150


1100








67

in the neon matrix. This makes intuitive sense since neon is the least perturbing of the

rare-gas matrices by virtue of the comparison of matrix shifted values for diatomic

ground state vibrational freqeuncies and electronic transitions.30'31 Again, there are

impurity peaks due to NbO, which in this case have been previously characterized by

Brom, Durham, and Weltner.37 They have assigned three absorptions due to sites in the

matrix for NbO at 965.5, 973.0, and 977.1 cm'. In our spectra, it seems that the only site

we see is the one at 973.6 cm'. This would not seem to be out of the realm of possibility

since the matrices were prepared in dissimilar manners.37 The peak at 989.4 cm1 that

appears in both spectra has not been assigned and probably is not due to NbO since these

peaks have been characterized previously, but it seems to be an impurity that cannot be

related to any Nb()C(n) product either. The transition at 914.5 cm"' could possibly be due

to NbO2 since only one site is seen here for NbO (in contrast to the argon data).

The Ground State (2Am) Vibrational Properties of Nb2C and Nb13C

Table 4.2 gives a summary of the ground state harmonic frequencies (O() and their

corresponding anharmonic corrections ((oXe) that were calculated from the

experimentally observed AG",,2 frequencies for the two isotopomers. The relationships

used in these calculations are given in equations 77 80.3" A comparison of the matrix

and gas phase values are given along with harmonic force constants.
12 12 12 2 1
AG1/2(Nb2C) = v2 = (02 2e,) e (77)
m e e12.,I

13 12
We = P(Oe (78)

13 13 2 12 12
We Xe P We Xe (79)










p = (1.2 / g 13)1/2 = 0.9652 (80)

The NbOmolecule

The fortuitous observation of NbO2 in an argon matrix has given us a chance to

speculate on the ground state of the NbO2 molecule. The ground electronic state of NbO2

has not been reported to our knowledge, but we would like to report the ground state

vibrational frequencies of v, and v3, and cautiously assign the v2 bending mode. The

symmetric stretch (v,) has three sites at 913.1, 907.2. and 901.1 cm' (see Figure 4.2). The

asymmetric stretch (v3) has three sites as well at 873.4. 867.8, and 859.6 cm'. The

bending mode would be expected to have a weak transition to the red of the above

absorptions. We have tentatively assigned the triplet around 518 cm' (524.5, 517.7, and

510.6) to v2 (see Figure 4.4).

With the apparent observation of a symmetric and asymmetric stretch, a bent

ground state seems to be in order. The bond angle of this ground state can be estimated

via the ratio of the relative intensities of V3 and v, with the assumptions made by Ozin et.

al.39 Figure 4.5 shows the enhanced region of the NbO, stretching region of Figure 4.2.

'v3 2 oM~b + 2McO sin 0181
V = tan2 0 ()
I t MNb + 2Mo cos2 )


The ratio of v3 and v, is based on the integrated intensities of the respective matrix sites.

The intensity ratio was determined to be 1.71, which gives a bond angle of 20 = 103".

This value agrees quite reasonably with Spiridonov et. al. (0 = 101.6(3.3)).40

Surprisingly, a theoretical treatment of NbO, has not been cited in the literature

tothe best of our knowledge. The most recent experimental work on NbO, was an













Table 4.1. Observed IR bands (cm') of niobium monocarbide and niobium dioxide in
rare gas matrices.

Molecule Neon Argon Krypton Assignment

Nb'2C 983.o 952.2 941.o AG",,2

Nb3C 949.7 917.7 908.5 AG",/2

NbO2 873.8, 867.8, v, (sym. str.)
859.6
NbO2 913.,, 907.,, v, (asym. str.)
901.1,
NbO, 524.5, 5177. v, (bending)


510.6


Table 4.2. Vibrational frequencies (cm"') of the ground state (2A3,2) of Nb'2C and Nb"3C
in rare-gas matrices at 4 K.


Species Parameter Ne

Nb2C AG",,2 983.0

o0e2 1013.o

12 12
We Xe 15.0

Nb'3C AG",, 949.7

1 3 975.2

S13 12.8

kb(mdyn/A) 6.42

'from reference 28.
calculated from relationship in reference 38.


Ar Kr Gas Phasea

952 0I1 QQf -4- 1 C


./V '-i- sJ


962.,

4.9

917.7

926.1

3.4

5.79


.*0

951.1

5.0

908.5

915.6

3.5

5.66































.0 0








0
0

0.












600 580 560







Figure 4.4. The three sites of the v2 bending mode of NbO2.


Wavenumbers (cm1)














R "








CDJ















9 0 910 900 890 880 810 860

Wavenumbers (cm-1)

Figure 4.5. The sites of the v, and V3 stretching modes of NbO2.








72

electron diffraction study performed by Spiridonov et al.40 They reported based on the

refinement of their data with v3 fixed at 1009 cm' that v, and v, were 854(41) and

527(41) cm' respectively. The values reported here are well within the error limits given

in the above study. Admittedly, they indicate that their treatment is very sensitive to

temperature, thus the reported vibrational frequencies have a parametric dependence on

temperature with a generous range of error 2678(39) K. In any event, the vibrational

frequencies determined in this experiment should give a clearer picture of the true

frequencies regardless of subtle matrix effects.

Conclusion

The experiment performed in krypton gives the best evidence that Nb2C and

Nb'3C were indeed synthesized. The isotopic shift in frequency ofNbVC (AG",, = 908.5

cm') relative to Nb2C (AG"1i = 941.0 cm') is quite satisfactory in comparison to a

calculated harmonic prediction (AG",a2 = 908.2 cm'). The krypton spectra (Figure 4.1)

clearly shows these isotopomers with very little interference from impurities. The

frequency of Nb'2C in the krypton matrix (AG"12 = 941.0o cm') has been substantially

reduced as compared to the gas phase value reported by Simard2l (AG"/112 = 980 15 cm').

The percent shift relative to the gas phase value is 3.9%. This is not atypical, as trapped

species in more polarizable matrices tend to have higher shifts relative to the gas phase.30

A comparison of other highly ionic species that have been trapped in matrices shows that,

for instance, LiF (dipole moment, ji = 6.33 D) has a shift in a krypton matrix of 6.9%

relative to its gas phase value.303' This would further validate the approximate magnitude

of the dipole moment that was calculated by Simard28 (g = 6.06 D).








73

The argon spectra (Figure 4.2) seem to exhibit bands of Nb2C and NbVC, but are

complicated by impurity peaks essentially due to NbO and NbO,. The isotopic shift in

frequency of NbC (AG",, = 917.7 cm-') relative to Nb'2C (AG",2 = 952.2 cm-') is again

quite good compared to the calculated harmonic prediction (AG",,2 = 919.0 cm'). The

percent shift of Nb'2C relative to the gas phase value is 2.8%.

The neon spectra (Figure 4.3) was the most speculative in terms of definitely

assigning bands to Nb'2C and Nb3C, which was discussed earlier. The bands that were

assigned to Nb3C (AG",, = 949.7 cm-') and Nb'2C (AG",2 = 982.0 cm-') agrees well with

the isotopic frequency compared to the calculated harmonic value (AG",,2 = 947.8 cm-').

The percent shift of Nb'2C relative to the gas phase value is -0.2%. This percentage is not

surprising since neon is the least perturbing matrix of the rare gases and most closely

resembles gas phase conditions.30













CHAPTER 5
DENSIMETER

Introduction

A complete understanding of the world that we "see" relies on the ability to

measure the behavior of a single atom as an isolated entity, as well as the collective

behavior of various ensembles of atoms and molecules that are placed under varied

environmental scenarios. Ongoing technological advances have enabled scientists to

actively research these extreme disciplines. For example, the ability to detect and

manipulate a single atom/molecule has recently been reported by several groups.41,42 At

the other extreme, condensed phase physicists have recently discovered the existence of

the Bose-Einstein condensate, predicted by Einstein nearly 70 years ago.43

The unusual behavior of a bulk state is not necessarily confined to the exotic

conditions of a Bose-Einstein condensate, but can also be realized in conditions of normal

fluids and mixtures that are far removed from the ambient. This behavior is particularly

evident near the critical point (Pa, pc, TJ), due to time-dependent fluctuations in density."44

As a result, these regions of a phase diagram can be difficult to define thermodynamically

due to deficiencies in thermophysical measurements. It has been pointed out by

Wagner20 that "the IUPAC Thermodynamic Tables Project Centre at Imperial College

London is committed to revise its tables on carbon dioxide,45 because the equation of

74








75

state used for establishing those tables show quite large systematic deviations from the

experimental (p, p, T) and caloric values, especially along the coexistence curve and in

the critical region."2 Thus, the need for reliable equations of state within the

experimental uncertainty of physical measurements is needed.

Furthermore, with the peaking interest of supercritical mixtures as a media for

novel chemical activity, the ability to predict the behavior of supercritical mixtures from

the complete and accurate thermodynamic surface of the media of interest is tantamount

to the complete exploitation of this area. These sentiments have been elucidated in the

literature by Poliakoff:46"'... reaction mixtures have to be homogeneous in order to exploit

the advantages of supercritical fluids. Therefore, a knowledge of their phase envelopes

and critical points is crucial. Even the simplest reaction system will be a ternary mixture

(reaction, product, and solvent).., and traditional methods of phase measurements (view-

cell or sampling) are often very difficult to apply to some ternary mixtures where the

density differences are small."46

The reoccurring theme of these observations, as stated in the literature, is the

ability to measure the state variables (i.e., p, p, and T) very accurately and with the

utmost precision. The state-of-the-art techniques for these measurements are adequate for

temperature and pressure, but density determinations are non-trivial, especially if a

system is well removed from the ambient. But, why is density so important?

Thermodynamic Relationships

The ability to control the behavior of bulk chemical systems involves using the

predictive power of a thermodynamic equation of state that is developed based on the








76

physical property measurements of these aforementioned independent variables. If these

variables are satisfactorily measured, all of the other thermodynamic properties of a fluid

can be expressed in terms of the Gibbs energy G (T,P) or in terms of the Helmholtz

energy A (T,p). It has been pointed out previously47 that for regions of low or moderate

pressures, either set of variables is fully satisfactory. However, when the two-phase,

vapor-liquid region is involved, the Helmholtz energy (A) is much more satisfactory

since it is a single valued function of T and p. This results from the density (p) being a

multi-valued function of temperature and pressure in this region of the phase diagram. It

is possible to tabulate or write separate equations for vapor and liquid properties in terms

of the Gibbs energy (G) (as functions of temperature and pressure) but, if one wishes a

single comprehensive equation of state, one uses T and p as variables and the Helmholtz

energy (A) as the parent function.

The parent function in both cases is the sum of two functions, one for the ideal-

gas or standard-state properties and the other for the departure of the fluid from ideal

behavior. Equations 81 and 82 show the Helmholtz energy (A) developed in terms of an

ideal gas and its relationship to its standard state properties, where pO = Po/RT (PO = 1 bar)

and Go = A + RT. The departure of the Helmholtz energy from the ideal case is related

by the following equations (83 and 84), where the compression factor z is defined as V. =

zRT/P' and P' = p'RT.

Aid(T, p) A(T) = RT ln(p / p0) (81)


= G(T) + RT[-1 + ln(pRT)]


(82)











id
[a(A- A ) / ap] = VRT- RT / p (83)
T m


P
p
id
A(T, p) A (T, p) = RTf (z l)d in p' (84)
0

As an aside, it should be mentioned that the development of an equation of state

is often given for the Helmholtz energy (A) instead of a compression factor (z), where

one first divides A into an ideal and a residual term (Equation 85). The compression


A = A id + Ares (85)

factor is then defined in terms of a residual Helmholtz energy (A'). But, if the

compression factor is initially defined as was the case here, the residual Helmholtz

equation (AS) is defined in equation 86, which essentially represents the departure of the

Helmholtz energy (A) from the ideal case (Aid). The final equation for the Helmholtz

energy is therefore equation 87. Substituting an appropriate equation of state for the


P
A / RT = J (z l)d in p' (86)
0

A(T, p) = Go + RT[-I + 1n(pRT)] + Ares (87)

compression factor (z-1) developed in terms of density, (the virial equation for example)

gives the complete Helmholtz energy (88), which can be utilized to derive most of the

other thermodynamic relationships.










A(T, p) = GOT) + RT[-1 + ln(pRTD + Bp + Cp2+. .] (88)

It is obvious that a nearly complete thermodynamic description of a normal fluid

can be obtained with the above relationships, but they are contingent on the accurate and

precise measurement of the state variables with density as the most elusive. With these

points in mind, the goal at the outset was the development of an all encompassing density

measuring instrument with the following criteria: 1) accuracy/preciseness, 2) sensitivity,

3) robustness, and 4) self-calibrating abilities (as an enhanced feature). Thus, the focus of

this study was the development of a state-of-the-art Archimedes type densimeter that in

principle would measure fluid densities over a wide range of temperatures and pressures.

First Principles

An Archimedes type densimeter relies on principles developed around 200 B.C.48

by one of the greatest scientists of antiquity, Archimedes. These principles are based on

the buoyancy force that is subjected to objects submerged in a fluid. For instance, the

forces that act on a rigid body that is immersed in a static, isothermal fluid are: 1)

Newton's second law of motion and 2) the buoyancy force (see Figure 5.1). The former

is the gravitational force (equation 89), where P is Newton's force of gravity, m is the

mass of the object, g is the acceleration of gravity, and i is a unit vector directed along

the center of the masses of the two bodies. The latter force (F) (equation 90) is

F =mgi (89)

dependent on the density (p) of the medium and the volume (V) of the fluid displaced by




















Fluid


Rigid
Sphere


re = pVgn


Br m gn


Figure 5.1. A schematic of the gravity and buoyancy forces on a sphere immersed in a static fluid.










the object. The local acceleration of gravity (g) and a are defined as they were in the

previous equation. These two forces act in opposition, and thus the resultant force


F = pVgn (90)
B

experienced by the immersed object is the intrinsic weight (F ) less the buoyancy
G

= (mg pVg)fi (91)
res

force ( ) (see equation 91). It is now evident that in principle it is possible to solve for
B

the density of an unknown fluid if the force of a submerged mass could be measured.

Design of Densimeter

The ability to measure the density of an object under ambient conditions is a

relatively simple matter. But, the sampling of density far removed from ambient

conditions is a non-trivial measurement. If one recognizes from equation 91 that the

density can be ascertained from the knowledge of the net force applied to an object in a

fluid, a density measurement would ostensibly be feasible. But, the net force on an

enclosed object under extreme pressures and temperatures is a very difficult

measurement. To begin to mitigate this problem, a spring can be used to suspend the

body within the fluid. Consequently, the force applied by the rigid body can be related to

the spring's displaced position. If the spring obeys Hookes' Law (which is an excellent

approximation over a very short deflection range), the deflection will be linearly








81

dependent on the total or net force placed upon the system (equation 92). In this

expression, k is the force constant of the spring and x is the displacement of the spring

from its equilibrium position.

F = -kxf (92)

If we consider the suspension of a sphere from a spring that has been described

above, the net force experienced by an individual sphere is now incident on the spring

and causes an expansion that is characteristic of the resultant load (equation 93). In this


F = (mg pVg)i = -kxd (93)
net

experiment, two rigid spheres (with masses m, and m2 and respective volumes V, and V2)

were placed in a rack and suspended from a spring in a static, isothermal fluid (see Figure

5.2). With this arrangement, it is possible to measure the force (specifically the position)

of four separate loads. Four linear equations are the result of the above measurement of

position as a function of the fluid's density (see equations 94 97). The displaced

FO = (mog pVog)i = -kxofi (94)

F = (mg- PVVg + mog pVog)fi = -kx i (95)

S= (m2g- pV2g + mog pVog)n = -kx2i (96)

F = (m3g pV3g + mog pVog)i = -kx3n (97)


position of the spring due to the net force of each loading was monitored by a

commercially available linear variable differential transformer (LVDT), which transduced






















































Figure 5.2. A schematic of spheres and a rack suspended from a spring.

00
t-J







83

the displacement to a voltage. The resultant equations that were used to solve for density

are represented below (equations 98 100), where S is the signal in terms of voltage and

K is the linear conversion of effective mass to voltage (At the outset, this constant is

assumed completely linear over the deflection range of interest.). The fourth

measurement (see equations 94-97) is a linear combination of the other three, thus there

F0 = (m0g pV0g)ii = -kx0 ii = S0 K(x) (98)

F = (m1g pVg + mog pV0 g)fi = -kx i = S K(x) (99)

F = (mg PVg + m g- pVg)i = -kx 2 i S K(x) (100)

are only three resultant equations with three unknowns ((m0-pV0), Kg/k, and p), which

can be solved to ascertain density. Note that the scaling constants (Kg/k) in these

equations are grouped together as one term.

The elegance of this particular design allows for the determination of the density

without the explicit knowledge of the local acceleration of gravity (g), the force constant

of the spring (k), or the LVDT constant (K). For example, the spring constant of the

material will vary with temperature and pressure, but it will not vary over a single density

measurement. Secondly, the local acceleration of gravity (g) should vary negligibly over

the distances (0.100") in this experiment. And, it assumed that a linear response will be

provided by the LVDT. Therefore. these constants cancel out of the resultant system mf

equations when they are solved And in theory, the instrument is a self-calibrating

device.










Densimeter Development

To bring to experimental fruition the highly attractive theoretical features of this

type of densimeter, an arduous trial and error procedure ensued. The critical component

and ultimately the foundation of this instrument was the sensing device that measured the

uniaxial displacement of the spring with a resultant load. As was mentioned in the

previous section, an extremely short deflection range was needed to ensure the linearity

of the deflected spring. Thus a sensing device was needed that had the combination of a

reasonably short stroke coupled with a high resolution of linear displacement. Therefore,

an LVDT (Linear Variable Differential Transformer) sensing device (Trans-Tek, Inc.

model #240-0015) was purchased with a deflection range of 0.100" and a resolution of

1 micron. The constituent parts of an LVDT consist of a weakly magnetic core (0.99"

OD X 0.492") and a cylindrical transformer with a cored center. The position of the

magnetic core relative to the transformer of the LVDT gives a linear voltage response

when it is displaced from a null position (a voltage of zero) to 6 volts

corresponding to 0.50" displaced distance.

With the desired specifications of the LVDT in hand, the next most important

aspect of the design and engineering process was to find a spring material that would lend

itself to the rigors of this experiment and more importantly have an extremely elastic

behavior over wide ranges of pressures and temperatures. In addition, the spring would

have a deflection of approximately 0.080" with a given load to take advantage of the

maximum stroke of the LVDT. The spring material that was chosen for use in this

apparatus was fused quartz. Quartz was chosen because of its extremely elastic behavior







85

under environmental stresses (a = 5.5e-7 cm cm"' C-'),49 and its resistance to hysteresis

when a load is placed on it over time (due to its amorphous structure). A stock piece of

fused quartz (0.06" OD) was heated, wound around a mandrel and placed in a lathe to

generate a coiled spring with the approximate dimensions of 0.25" OD and 0.5" Length.

Excellent control of the spring constant could be exercised with bench top measurements

of deflection by etching the quartz (2.5 M) over carefully monitored time periods to

obtain the desired deflection of approximately 0.80" with a load of 25g.

From a previous design of another densimeter model.50 two hollow stainless steel

spheres were fabricated and used as a changeable load that was to be placed on a rack. In

the current design, the rack and the spheres would then be suspended from a spring and

the displacement sensed by the LVDT. The sinkers were originally designed to have

approximately the same volume (V,=V2), but one sinker would have twice the mass

(2m,=m,). They were machined from hemispheres made of 304 stainless steel with a

0.625"OD and a 0.437" ID, which gave a sufficient wall thickness to prevent any

dimensional change (and therefore volume change) up to 3000 psia. A copper bead that

had the mass of two hemispheres was placed in one set of hemispheres (to achieve

approximately twice the mass of the other), and the two sets of hemispheres were then

Heli-Arc welded at the seam. The masses and volumes of the spheres were determined

via an analytical balance and pycnometric volume determinations.50

The remaining parts of the densimeter were built around the specifications of

these integral components. To begin with, a rack to hold the buoys was fabricated to








86

suspend from the quartz spring. The rack was designed to incorporate the LVDT

magnetic core into a suspension post that screwed into the rack so as to make a single unit

(see Figure 5.3 for schematic). This rack was assembled by brazing two 0.062" diameter

stainless steel rods onto a brass disc. The actual supports for the spheres were short

pieces of stainless steel wire that were brazed in a V-shape to these rods. The brass disc

was drilled an tapped 5-40. Two brass posts were then fabricated to connect the brass

disc to the quartz spring, and linked between them was the magnetic core of the LVDT.

The first rod was threaded 5-40 on one end to place into the brass disc, and the other end

was threaded 1-72 to place into the LVDT magnetic core. The second brass rod had a

small hole drilled in the center of a small flattened section to accommodate a hook that

extended from the quartz spring to link the rack to the spring. And, finally the other end

of this second rod was threaded 1-72 on one end (to place into the other end of the

magnetic core).

With the rack unit in place, there was the need to independently lift the buoys

from the rack. Mechanical lifters were designed to fit into a cylindrical, brass sleeve

(1.049" OD), which in turn would slip fit around the rack and spheres. Vertical slides

were dimensioned to provide proper clearance for each individual loading so as to

prevent any mechanical contact between the rack and the sleeve and to ensure that the

load would hang freely from the spring. The lifters were actuated via a magnetically

coupled cam system that lifted the spheres from the rack to provide the instrument with

four independent loads.
























































Figure 5.3. A schematic of the LVDT core mounted between two brass posts.
00
-.4








88

The aforementioned brass sleeve (Figures 5.4 5.9) was machined with two slots

that were 900 apart to accommodate the lifter slides. These slides were made from brass

(3.990" x 0.185" x 0.145") and were fitted with stainless steel forks (1/32" diameter) that

were brazed into the body of the lifter slides. The forks had three points that provided

support for spheres when lifted from the rack. Finally, the lifter slides were slotted at the

bottom and fitted with small brass wheels to ride on the cam assembly to actuate the

motion of the lifter slides (see Figure 5.10). The magnetically driven cam assembly

(Figure 5.11) is a two-tier cam equipped with an internal magnet, which is coupled to an

external magnet. The cam has a 450 step with a total rise of 0.250". The quadrants of the

cam allow for the motion of the lifters to give four separate loads referred to earlier. The

cam is fitted onto a magnetic turner (Figure 5.12) by a stem that locked into position by

a set screw. The cam rides on brass ball bearings (Figure 5.13) placed in between the

cam and the magnetic holder. In turn, the cam and magnetic turner unit rotates on a set of

brass ball bearings that sets on the bottom of the densimeter can. A ferromagnet was

placed into a drilled out section of the magnetic turner, which was held in place with a set

screw. The ferromagnet was coupled to an external magnet that drove the cam assembly.

At this point, the densimeter was assembled into two main components (see

Figure 5.14 for schematic assembly drawing). The upper component consisted of the

following pieces: 1) a top flange (Figure 5.15) equipped with a brazed spring mount, a

ball bearing support clamp for a drive screw (see Figure 5.16), a drive screw (Figure

5.17) that actuated the movement of the LVDT housing (Figure 5.18 and 5.19), and a top










flange cover (Figure 5.20). 2) a bottom flange (Figure 5.21) that was married to a

complimentary flange on the densimeter body and supported the top flange via support

legs (Figure 5.22). The top and bottom flanges were physically joined by an inconel'

sleeve (see Figure 5.23) that traversed the LVDT housing and encompassed the LVDT

core.

The lower component was made up of a densimeter body (Figure 5.24) that

contained an end cap (Figure 5.25) brazed into the bottom, a magnetic turning unit, and a

cam assembly. A flange (shown in Figure 5.24) was brazed onto the densimeter body

(complimentary to the bottom flange) in order to marry the two components together and

create a high pressure seal. This seal utilized a copper gasket that was fabricated by

cutting a piece of 12 gage copper wire to a length of 3.92", and the ends Heli-Arc welded

together to make a continuous piece. The copper gasket was implemented by initially

placing it around the diameter of the densimeter body and resting it on the offset of the

bottom flange. The upper was then placed on top of the bottom flange and the twelve

10-32 bolts were tightened to compress the copper wire into the gland between the two

flanges. The gasket is dimensioned to fill 95% to 98% of the triangular cross-sectional

area of the gland.

Experimental

To initially test the viability of this instrument; temperature, pressure, and density

data for argon and carbon dioxide were acquired over several isotherms near ambient

temperature, but over a wide range of pressures and thus density. These tests would bring



















WW AON Si a .74 dp


.-M dam, as d.p
a .ai *M


I- fM -1


to ," W
*, l.I


Figure 5.4. The cut-away view of the brass sleeve.


I a


















12:00 and 6:00 position

-4 .a." o-



S I Ii
I.OU








*H IM* ^
IN ~* -~!


K- LOM


Figure 5.5. The 12:00 and 6:00 position of the brass sleeve.






12.:00


9.:00


POSITION


1.30


:00


4:30


6:00


Figure 5.6. The respective positions of the brass sleeve.







93

1: 30 position









.0Z d9P
\W Id.
--- -4 ,.
________________ r- *>** wUi


^-^ .^ "* ,


.O w s a .T deep


I- .I-


Figure 5.7. The 1:30 position of the brass sleeve.





3*.00


+


9:00


position


Figure 5.8. The 3:00 and 9:00 positions of the brass sleeve.


file top corners
of slot square


6:00